Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.preservesSheafification_iff_of_adjunctions

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] (J : CategoryTheory.GrothendieckTopology C) {A : Type u_1}
  {B : Type u_2} [inst_1 : CategoryTheory.Category.{v_1, u_1} A] [inst_2 : CategoryTheory.Category.{v_2, u_2} B]
  (F : CategoryTheory.Functor A B)
  {G₁ : CategoryTheory.Functor (CategoryTheory.Functor Cᵒᵖ A) (CategoryTheory.Sheaf J A)}
  (adj₁ : G₁ ⊣ CategoryTheory.sheafToPresheaf J A)
  {G₂ : CategoryTheory.Functor (CategoryTheory.Functor Cᵒᵖ B) (CategoryTheory.Sheaf J B)}
  (adj₂ : G₂ ⊣ CategoryTheory.sheafToPresheaf J B),
  J.PreservesSheafification F ↔
    ∀ (P : CategoryTheory.Functor Cᵒᵖ A),
      CategoryTheory.IsIso (G₂.map (CategoryTheory.Functor.whiskerRight (adj₁.unit.app P) F))

Defined in

Mathlib.CategoryTheory.Sites.PreservesSheafification

Cited by

1 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category